On the slopes of phase boundaries
- PDF / 482,210 Bytes
- 7 Pages / 594 x 774 pts Page_size
- 35 Downloads / 221 Views
I.
INTRODUCTION
PHASE diagram compilation and evaluation is a topic of great current interest. The alloy phase diagram compilation program of the American Society for Metals has been underway for several years. The American Ceramic Society is lauching a similar project for ceramic phase diagrams. Thousands of diagrams are being evaluated under these international programs9 In evaluating a phase diagram, it is important to check that the diagram is consistent with the thermodynamic properties of the system. Ideally, a complete computerassisted optimization of all available thermodynamic and phase equilibrium data on a system should be performed with a view to obtaining equations for the Gibbs energies of the phases which can be used to calculate the phase diagram and the thermodynamic properties in a completely self-consistent manner. Many systems have already been fully optimized in this way. However, there are also a number of simple equations which can be used for providing a quick check on the consistency of phase diagrams. These need only a hand calculator for their solution and often require no more thermodynamic data than the entropies of fusion of the components. The best example of such an equation, which has been known for a century, but which is still not used regularly, relates the limiting slopes of the liquidus and solidus in a binary system when the mole fraction of one component is unity:
(dr/dX~);la=, _ (dT/dXSA);•=,_
=
o o 2 Ahf(A)/R(T~A))
II.
RATIOS OF SLOPES AT INVARIANTS
In Figures 3(a) through (d) are shown four cases of invariants involving three phases: a,/3, and 3~in a binary system with components A and B. Let 0% and o-~ be the slopes of the "y-phase boundaries of the (3/+ cz) and (3/+/3) regions at the invariant temperature as shown in the figure:
o% = (dT/dXB)~,,~
0"3"~ = (dT/dXB)3"t~
0"3,t8 :
[ X Aa( S ay -
S~t ) -3L X Ba( S B3' -- SB) ] [ X ~ x
(s,
~
- 40
~176176
K
02
04
XNa
0.6
[3]
/ t
J, I,
l:
X BT]
sg)] [xT, -
-
[1]
ARTHUR D. PELTON is Co-Director, Centre for Research in Computational Thermochemistry, Ecole Polytechnique de Montreal, P.O. Box 6079, Station A, Montreal, PQ, Canada H3C 3A7. Manuscript submitted August 12, 1987.
[2]
where XB is the mole fraction of B. The following completely general expression for the ratio 0"3'J0% is derived in the Appendix by applying the GibbsDuhem and Gibbs-Helmholtz equations. The derivation involves no assumptions.
-
where (dT/dXLa),A=~ and (dT/dXS),A=~ are the slopes of the liquidus and solidus when the mole fraction, XA, of component A equals unity, Ah:~a) is the molar enthalpy of fusion of A, and T~,) is the melting point of A (in kelvins). The only requirement involved in Eq. [1] is that Raoult's law be obeyed in the limit for the solid and liquid phases. In Figure 1 is shown the K-Na phase diagram. Itl The experimental points are from Reference 2. The experimental limiting liquidus and solidus slopes at XK = 1 are drawn on the diagram. These are read off the diagram as (dT/dX~c) = 270 and (dT/dX s) = 1120.
Data Loading...
